As one of my heroes said in a debate, "What an incredibly stupid question."
The reason I say this is that the answer leads nowhere. Say the answer was yes. Then what? No? Who cares? After the question has been answered you're exactly where you left off. So there is little point in asking to begin with. If you want a short answer, it's no, but the real fun is in arguing about why so here goes.

I think the question is flawed. The definition of art is incredibly wishy washy to begin with. You can say generally that it is based on aesthetic principles; what is beautiful, appealing, or meaningful in some way. You can also define art by fiat, by making a list of media which are artistic (painting, sculpture, theatre, ...).

Maskelyne described three levels of art in Our Magic (with David Devant, 1911).

Of course in his introduction to Our Magic, Maskelyne presupposed that magic was an art and that no one would dare argue that point. Most people reserve the word art for High Art and refer to the other two as craft.

The other problem is that art cannot exist independently of its audience. If enough people, or certain authoritative people, declare something is artistic, then it is. This reduces art largely to a contest of popular opinion. And it will always It's also a definition that can only come from the outside. It's like being a lady: if you have to declare you're a lady, you probably aren't.The same goes for artists.

How does magic stack up with these definitions? Badly:

If the classes of art are pre-defined based on their medium, the only hope magic has is to sneak its way in as a subclass of theatre (Doug Henning, the only magician to ever extract arts funding from the Canadian Government did so using the slogan "Magic + Theatre = Art"). I believe this distinction is an artifact of history. Magic came into its own relatively recently. Painting, sculpture, and possibly theatre have a head start of a few thousand years. Magic also spent a long time hiding within the realm of shamanism and the occult. (If you go see a witchdoctor, and find yourself feeling better because the placebo effect, this is a kind of magic.) It has only been in the past few centuries that magic could be said to exist as an independent field.

For those looking to quantify that number, there are a few salient dates. In 1584, Reginald Scott wrote Discoverie of Witchcraftwhich was aimed at convincing people that witchcraft was not real (so you didn't have to go around hunting them) and included explanations for some magic stunts. There are other publications in French, Spanish and Italian which are contemporary +/- 100 years which have magic lumped in with physical and mathematical amusements (the equivalent of a making baking soda volcanoes). Explicitly non-occult performance magic emerged in the 1800s (the French term translates literally as white magic) and was popularized by Robert-Houdin).

Perhaps magic could establish itself based on its aesthetic appeal. I doubt this can happen, since most of the technique for magic is hidden. So the only people who would be qualified to properly evaluate the performance would be those who already know how the tricks work. This is contrary to most other arts where technique is largely on display. And we compound the problem by lying about our methods. Think of Robert-Houdin employing ether fumes to levitate his son, or present day mentalists who claim to be using NLP.

And what if we win popular consensus? Things certainly do appear to be moving in the right direction. Select universities are adding magic courses, festivals are accepting magic shows and scientists are taking an interest in magic for research purposes. But it's too soon to tell if this progress will continue or if magic will max out in the near future.

So why is this so important to magicians? Perhaps they enjoy the sound of their own arguments. Of course the main obsession is the privileged status artists receive when it comes to asking people for money. If you can get yourself declared an artist, you can try to squeeze free money out of various arts organizations. The motivation seems a bit transparent. So much for being lady like.